A secret trick in the Math Olympiad; and mechanical drawing...

I did this drawing fully by myself. It's only meant to indicate the
8 bores and 18 bolt holes, though...

Getting the bolt holes to 'dodge' the inlet ports was a tricky effort. I used a sequence to generate the "numerical positions" (0 to 360) of the bolt holes, arbitrarily starting at the north cardinal direction. Then I wrote a separate column to indicate the axial (through the centre of the cylinder to the circumference) positions of the inlet ports, and slotted arrows inbetween the numbers representing the bolt holes.

It sounds nice, but it was really rough work.

Anyway, by rotating the bolt holes to begin 5 degrees clockwise from the vacuum pump inlet port, and rotating the view port by 10 degrees clockwise, the bolt holes have a minimum angular separation of 5 degrees from the ports. I was worried that the bolts wouldn't fit, so I used a scale drawing and found the minimum clearance (distance between the radius of bolt circle and a 1" rectangle representing the largest linear dimensions of the inlet ports) to be 2mm on paper: 6mm in person.

Voila. 30 seconds effort.

Which brings me back to a [week] old story.

While I was supervising (all right - lazily watching) Ryan at work on the mechanical drawings, I learned that he liked to use the empirical method of finding dimensions. For example, the electrodes don't form perfect semicircles: I use Pythagoras' theorem twice to correct the 'pseudo-diameter' from the diameter. Or the bolt centres on the electrodes needed reference dimensions, so I would start reciting "10/2, 10, 3, 10, 15/2...." and sum them up... (On which matter, I feel that I've become lazier nowadays - a visual image didn't come up in my head, when it really should: it would have been faster.)

On the other hand, Ryan would take his scale drawing from another sheet of paper (base plate), and move it over (the electrode drawing), and try to mark out the dimensions from there.

He got 106mm - which was wrong. He later (a few days) corrected it to 56mm - which was wrong. I found the separation to be 65mm.

So I realized that I like dealing with theoretical calculations more than empirical measurements. But it's a matter of effectiveness. I decided to use an empirical measurement to find the 6mm clearance calculation because it's the faster method in this case. And it didn't matter what the exact clearance was so long as it could accommodate the largest radius in the mounting hardware (the flat washers).

Which brings me to a story. I was on the same team as the Vietnam national youth champion (or second place - can't remember) in chess. I decided to introduce him to a game called Sprouts because it is just awesome (the world would be a better place if everyone knew how to play it - then you wouldn't have nothing to do so long as you had pen and paper, and someone else besides you who has nothing to do). And we talked. I found out that he was also representing the school in the Math olympiad.

As we were discussing the Math olympiad, I told him that the questions on inequalities were the hardest for me. Number theory is naturally difficult because it isn't taught in the A-Level syllabus, but not as difficult as inequalities. Functional equations, algebra, geometry etc. were OK, but I felt that geometry required some rote memory.

Then, he said, "I never need to study for geometry."

I was surprised.

"Wow, you must be really good."

"Nooo, I teach you a secret trick... you bring a compass and protractor. The problems are always drawn-to-scale, so I always draw them and measure and I get all of them correct. Sometimes you see 58, 59, you must see... and maybe round up to 60."

I laughed, "I thought protractors weren't allowed in the Olympiad?"

"They don't care. No one is watching."

We talked more, and I could tell that he knew his stuff and could write the proofs for the inequalities questions better than I could. I learned a lot from him - and I'm always glad to meet people whom I have much to learn from.

When someone told me that he thought I was the strongest math student in school just that I have a lot of involvements that I don't have the time to inflate my scores, I told him that the latter was true, but I definitely didn't have the strongest math in the school. I doubted that there was such a thing as "strongest math" unless you are distinctly overpowered (like my fluids research mentor, who already knew the answer to a calculation before he finished verbally reading out the expression that was to be evaluated) and I'm sure there were many who were more deserving of that position than me.

So, I told him, "There's this guy in our level who is many times better than me at math." I really think so.

I probably wouldn't have used his "secret trick" to find the minimum clearance if I never met him, because I would never trust empirical measurements. I always preferred theoretical research to experimental research. But it was a secret trick worth learning (but I didn't use it during the Olympiad of course - I still felt that it was against my morals).

But you know what's the real secret trick? Finding people whom you enjoy learning math with.